Subgroups — #27538757

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20276880	C₂	1	4	yes	maximal, complemented
4	#20276868	C₂	1	2	yes	maximal, complemented
5	#20276858	C₂	1	2	yes	maximal, complemented
6	#20276503	C₂	1	2	yes	maximal
7	#20276436	C₂	1	2	yes	maximal
8	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	#16811180	—	2	1	no	—
11	C₂×C₄×Dic₁₀₆	—	2	1	no	—
12	#16811028	C₂²	1	1	yes	characteristic
13	#16811028	C₂²	1	1	yes	characteristic
14	#16811025	—	2	1	no	—
15	#16811025	C₂²	1	2	yes	—
16	#16811021	—	2	1	no	—
17	C₄×Dic₂₁₂	—	2	1	no	—
18	C₄×Dic₂₁₂	—	2	1	no	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	C₄×Dic₂₁₂	C₂²	1	2	yes	—
21	#16810661	—	2	1	no	—
22	#16810661	C₂²	1	2	yes	—
23	C₂×#13133151	C₂²	1	1	yes	characteristic
24	C₂×#13133142	—	2	1	no	—
25	#16810451	C₂²	1	4	yes	—
26	#16810449	C₂²	1	4	yes	—
27	#16810428	C₂²	1	4	yes	—
28	#16810428	C₂²	1	4	yes	—
29	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₂²×Dic₁₀₆	—	2	1	no	—
34	C₂×Dic₂₁₂	—	2	1	no	—
35	C₂×Dic₂₁₂	—	2	2	no	—
36	C₂×Dic₂₁₂	Q₈	1	4	yes	complemented
37	C₂×Dic₂₁₂	C₂³	1	2	yes	—
38	#13133217	—	2	1	no	—
39	#13133217	—	2	1	no	—
40	#13133209	—	2	2	no	—
41	#13133196	—	4	1	no	—
42	#13133196	—	4	1	no	—
43	C₄×Dic₁₀₆	—	2	2	no	—
44	C₄×Dic₁₀₆	—	2	2	no	—
45	C₄×Dic₁₀₆	—	4	1	no	—
46	C₄×Dic₁₀₆	—	4	1	no	—
47	C₄×Dic₁₀₆	—	4	1	no	—
48	C₄×Dic₁₀₆	—	4	1	no	—
49	#13133176	—	2	2	no	—
50	C₂×C₄⋊Dic₅₃	—	2	1	no	—
51	C₂×#11719919	—	4	1	no	—
52	C₂×C₄×Dic₅₃	—	4	1	no	—
53	#13133151	—	2	1	no	—
54	#13133151	—	2	1	no	—
55	#13133151	—	2	1	no	—
56	#13133151	C₂³	1	2	yes	—
57	#13133151	C₂³	1	2	yes	—
58	#13133151	C₂³	1	2	yes	—
59	#13133149	—	2	2	no	—
60	#13133149	—	2	2	no	—
61	#13133142	—	2	2	no	—
62	#13133142	—	2	2	no	—
63	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
66	M₃₂×C₅₃	—	2	4	no	—
67	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
68	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₅₃	—	4	1	no	—
76	C₂²×Dic₅₃	—	4	1	no	—
77	C₂×Dic₁₀₆	—	2	4	no	—
78	C₂×Dic₁₀₆	—	2	2	no	—
79	C₂×Dic₁₀₆	—	4	1	no	—
80	C₂×Dic₁₀₆	—	4	2	no	—
81	C₂×Dic₁₀₆	—	4	1	no	—
82	C₂×Dic₁₀₆	—	4	1	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₅₃	#41	1	2	yes	—
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₅₃	D₈	1	1	yes	characteristic
91	C₄²×C₅₃	D₈	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²⋊Dic₅₃	—	4	2	no	—
95	C₄⋊Dic₅₃	—	2	2	no	—
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	—	2	2	no	—
98	C₄⋊Dic₅₃	—	4	1	no	—
99	C₄⋊Dic₅₃	—	4	1	no	—
100	C₄⋊Dic₅₃	—	4	1	no	—
101	#11719919	—	4	2	no	—
102	#11719919	—	4	2	no	—
103	#11719919	—	4	2	no	—
104	#11719919	—	4	2	no	—
105	C₄×Dic₅₃	—	4	2	no	—
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	—	4	2	no	—
108	Dic₂₁₂	—	2	4	no	—
109	Dic₂₁₂	—	2	2	no	—
110	Dic₂₁₂	C₂×Q₈	1	4	yes	—
111	C₁₆×C₅₃	—	2	4	no	—
112	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#137	1	2	yes	—
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₅₃	—	4	2	no	—
124	C₂×Dic₅₃	—	4	4	no	—
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	—	4	2	no	—
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	8	2	no	—
129	Dic₁₀₆	—	2	4	no	—
130	Dic₁₀₆	—	4	2	no	—
131	Dic₁₀₆	—	4	4	no	—
132	Dic₁₀₆	—	8	2	no	—
133	Dic₁₀₆	—	8	2	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
137	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
139	#33628	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	#474	1	2	yes	—
143	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	#474	1	2	yes	—
148	C₄×C₅₃	#510	1	2	yes	—
149	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
151	Dic₅₃	—	4	4	no	—
152	Dic₅₃	—	8	2	no	—
153	Dic₅₃	—	8	2	no	—
154	Dic₅₃	—	8	2	no	—
155	#3160	—	53	1	no	—
156	C₂×C₄×Q₁₆	—	53	1	no	—
157	#2229	—	53	4	no	—
158	#2227	—	53	2	no	—
159	#2225	—	53	2	no	—
160	#2159	—	53	2	no	—
161	#2150	D₅₃	1	1	yes	characteristic
162	#2141	—	53	2	no	—
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
165	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
166	C₂×C₅₃	#3414	1	1	yes	characteristic
167	C₂²×Q₁₆	—	53	1	no	—
168	#533	—	106	1	no	—
169	C₂×C₄×Q₈	—	106	1	no	—
170	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
171	#500	—	53	1	no	—
172	#500	—	53	1	no	—
173	#499	—	53	2	no	—
174	#499	—	106	1	no	—
175	#498	—	106	1	no	—
176	C₄×Q₁₆	—	53	2	no	—
177	C₄×Q₁₆	—	53	2	no	—
178	C₄×Q₁₆	—	106	1	no	—
179	C₄×Q₁₆	—	106	1	no	—
180	#427	—	53	2	no	—
181	#427	—	106	1	no	—
182	C₂×#107	—	53	1	no	—
183	C₂×#103	—	106	1	no	—
184	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
185	#367	—	53	4	no	—
186	#366	—	53	4	no	—
187	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
188	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
189	#358	—	53	4	no	—
190	#358	—	53	4	no	—
191	C₅₃	#33628	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	106	1	no	—
193	C₂×Q₁₆	—	53	4	no	—
194	C₂×Q₁₆	—	53	2	no	—
195	C₂×Q₁₆	—	106	1	no	—
196	C₂×Q₁₆	—	106	2	no	—
197	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₄⋊Q₈	—	106	1	no	—
200	C₄⋊Q₈	—	106	1	no	—
201	#125	—	106	2	no	—
202	C₂²⋊Q₈	—	212	1	no	—
203	C₂²⋊Q₈	—	212	1	no	—
204	C₄×Q₈	—	106	2	no	—
205	C₄×Q₈	—	106	2	no	—
206	C₄×Q₈	—	212	1	no	—
207	C₄×Q₈	—	212	1	no	—
208	C₄×Q₈	—	212	1	no	—
209	C₄×Q₈	—	212	1	no	—
210	#117	—	106	2	no	—
211	C₂×C₄⋊C₄	—	106	1	no	—
212	C₂×C₄⋊C₄	—	212	1	no	—
213	C₂×C₄²	—	212	1	no	—
214	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
215	M₃₂	—	2	4	no	—
216	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
217	#107	—	53	2	no	—
218	#107	—	53	2	no	—
219	#107	—	53	2	no	—
220	#107	—	106	1	no	—
221	#107	—	106	1	no	—
222	#107	—	106	1	no	—
223	#106	—	106	2	no	—
224	#106	—	106	2	no	—
225	#103	—	106	2	no	—
226	#103	—	106	2	no	—
227	C₄×C₈	D₂₁₂	1	1	yes	characteristic
228	C₄×C₈	D₂₁₂	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
231	C₂×Q₈	—	106	4	no	—
232	C₂×Q₈	—	106	2	no	—
233	C₂×Q₈	—	212	1	no	—
234	C₂×Q₈	—	212	1	no	—
235	C₂×Q₈	—	212	1	no	—
236	C₂×Q₈	—	212	2	no	—
237	C₂²×C₄	—	212	1	no	—
238	C₂²×C₄	—	212	1	no	—
239	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
240	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
242	Q₁₆	—	53	4	no	—
243	Q₁₆	—	106	4	no	—
244	Q₁₆	—	106	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
247	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
248	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
249	C₂×C₈	#11719947	1	2	yes	—
250	C₂×C₈	Q₈×D₅₃	1	2	yes	—
251	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	106	2	no	—
253	C₄⋊C₄	—	106	2	no	—
254	C₄⋊C₄	—	106	2	no	—
255	C₄⋊C₄	—	212	1	no	—
256	C₄⋊C₄	—	212	2	no	—
257	C₄⋊C₄	—	212	2	no	—
258	C₄⋊C₄	—	212	2	no	—
259	C₄⋊C₄	—	212	2	no	—
260	C₄⋊C₄	—	212	1	no	—
261	C₄⋊C₄	—	212	1	no	—
262	C₂²⋊C₄	—	212	2	no	—
263	C₄²	—	212	2	no	—
264	C₄²	—	212	2	no	—
265	C₄²	—	212	2	no	—
266	C₄²	D₄₂₄	1	1	yes	characteristic
267	C₄²	D₄₂₄	1	1	yes	characteristic
268	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
269	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
270	C₁₆	—	2	4	no	—
271	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
272	Q₈	—	106	4	no	—
273	Q₈	—	424	2	no	—
274	Q₈	—	424	2	no	—
275	Q₈	—	212	4	no	—
276	Q₈	—	212	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	424	2	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	—	212	2	no	—
283	C₂×C₄	—	212	2	no	—
284	C₂×C₄	—	212	4	no	—
285	C₂×C₄	#13133197	1	2	yes	—
286	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
291	C₂×C₄	#13133250	1	2	yes	—
292	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#13133197	1	2	yes	—
296	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
297	C₈	C₂×#11719947	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#16810904	1	2	yes	—
301	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	424	2	no	—
306	C₄	—	424	2	no	—
307	C₄	—	424	2	no	—
308	C₄	—	212	4	no	—
309	C₄	#16810904	1	2	yes	—
310	C₄	#16811066	1	2	yes	—
311	C₄	C₂×#13133197	1	1	yes	characteristic
312	C₄	C₂×#13133250	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₂×#16810904	1	1	yes	characteristic
315	C₂	C₂×#16811066	1	1	yes	characteristic
316	C₂	#20287851	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#33628
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20276880	C₂	semidirect
4. #20276868	C₂	semidirect
5. #20276858	C₂	semidirect
36. C₂×Dic₂₁₂	Q₈	semidirect
191. C₅₃	#33628	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20285229	2
C₂×C₄×Dic₂₁₂	2
#20276880	2
#20276868	2
#20276858	2
#20276503	2
#20276436	2
C₅₃×#2150	2
#33628	53